Multidiameters and Multiplicities

The k-diameter of a graph 0 is the largest pairwise minimum distance of a set of k vertices in 0, i.e., the best possible distance of a code of size k in 0. A k-diameter for some k is called a multidiameter of the graph. We study the function N (k,1, D), the largest size of a graph of degree at most1 and kdiameter D. The graphical analogues of the Gilbert bound and the Hamming bound in coding theory are derived. Constructions of large graphs with given degree and k-diameter are given. Eigenvalue upper bounds are obtained. By combining sphere packing arguments and eigenvalue bounds, new lower bounds on spectral multiplicity are derived. A bound on the error coefficient of a binary code is given.